Continuous Inverse Theory and Tomography

Menke, William

doi:10.1016/b978-0-12-397160-9.00011-4

Cited by 128 publications

(196 citation statements)

References 6 publications

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“…When C d is I, which is the case in this particular example, equation (9) is also known as Tikhonov regularization (Tikhonov, 1977) and damped leased squares (Menke, 2012).…”

Section: Resultsmentioning

confidence: 96%

Localized Smart Interpretation - a data driven semi-automatic geological modelling method

Lundh

Torben²,

Bach

et al. 2015

ASEG Extended Abstracts

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INTRODUCTIONWhen setting up geological models today, an enormous amount of information may be available, e.g. airborne electromagnetic data (AEM-data), borehole data, radiometric data, seismic data, geological information, etc. However, it is only possible to incorporate a limited amount of the accessible information in the geological modelling process due to the very time demanding task of manual interpretation work. We suggest a methodology to target this problem, which infers a linear model that describes the relation between geological interpretation and the information available to the geologist making the interpretation. Once such a model has been inferred, it can be used, in a semi-automatic way, to perform new (i.e., predict) geological interpretation based on and consistent with the interpretation made by the geological expert. METHODIdeally the method constructed will build a statistical model f (d,M), describing the relation between what the geologist interprets, d=(d 1, d 2, …, d M ) T , and the quantifiable information (i.e., attributes) available to the geologist, M = (m 1, m 2, …, m N ). Each vector m i contains the i'th attribute, which can be any available information that can be quantified, such as geophysical data, the result of geophysical inversion, elevation maps etc. The vector d contains actual interpretations, such as for example the depth to the base of a ground water reservoir. First, the statistical model f(d,M) should be inferred by examining sets of actual interpretations made by a geological expert d and the attributes used to perform the interpretation M. This makes it possible to formulate a probability distribution f(d,M) that describes the relation between attributes and the geological interpretation. As the geological expert proceeds interpreting, the number of interpreted data points from which the statistical model is inferred, increases and, consequently, the accuracy of the statistical model increases. When a model f(d,M) has been successfully inferred, it is possible to predict how the geological expert would perform an interpretation d pred given some external information M ext through f(d pred |M ext ).In this paper we assume a linear relation between d pred and M ext such that:and seek to find the d pred , which gives the highest probability value of the conditional probability distribution (d pred |M ext ). In equation (1) SUMMARYLocalised Smart interpretation (LSI) is a method that infers a statistical model, which describes a relation between the knowledge of a geologist (as quantified by geological interpretation) and the available information (such as geophysical data, well log data, etc.) that a geologist uses when he/she interprets. This model is then used to perform semi-automatic geological interpretation wherever the same kinds of attributes, as used for the initial interpretation, are available. The statistical model is inferred using a combination of a regularized least squares method and cross validation. In this study, we demonstrate the applicability of...

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“…When C d is I, which is the case in this particular example, equation (9) is also known as Tikhonov regularization (Tikhonov, 1977) and damped leased squares (Menke, 2012).…”

Section: Resultsmentioning

confidence: 96%

Localized Smart Interpretation - a data driven semi-automatic geological modelling method

Lundh

Torben²,

Bach

et al. 2015

ASEG Extended Abstracts

View full text Add to dashboard Cite

show abstract

“…It is also helpful if rigid body modes are removed from stiffness matrices [14], which is not an imperative with a good algorithm for generalized matrix inverse (e.g. with application of singular value decomposition if necessary, see [15]). In Fig.…”

Section: Example 2: Similarity For Replacing Beam Part With Equivalenmentioning

confidence: 99%

Similarity of structures based on matrix similarity

2017

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Original scientific paper The paper presents a numerical procedure for relating the behaviour of two different structures, i.e. determining a scale between two structures. This novel solution is based on the notion of matrix similarity and linear transformations, with the restriction that the scale between structures is determined only after structural discretization, and that both structures have to be in the elastic regime. The structure scale can be determined in loading space or displacement space (i.e. structure forces or displacements are put into relation) where the scaling of the static structure model is based on the matrix equivalence principle, and scaling of the dynamic structure model is based on the Smith normal form. The structure scale in operator space (structure stiffness or flexibility matrices are put into relation) should be based on the Sylvester matrix equation. However, that approach is not practical and is replaced with the Levenberg-Marquardt method for obtaining only approximately equivalent stiffness matrices. Numerical examples illustrate the proposed novel approach.Keywords: Levenberg-Marquardt method; matrix equivalence; similarity of structures; Smith normal form; structure scales; Sylvester equation Sličnost konstrukcija zasnovana na sličnosti matricaIzvorni znanstveni članak Rad prikazuje numerički postupak za uspostavljanje odnosa između ponašanja dvije različite konstrukcije, odnosno određivanje mjerila (faktora skaliranja) između dvije konstrukcije. Ovo novo rješenje zasnovano je na ideji sličnosti matrica i linearnim transformacijama, uz ograničenja da se mjerilo između konstrukcija određuje tek nakon provođenja diskretizacije te da obje konstrukcije moraju biti u elastičnom području. Mjerilo konstrukcije može se odrediti u polju opterećenja ili pomaka (ovisno o tome dovode li se u vezu sile ili pomaci konstrukcije) gdje se skaliranje statičkog modela konstrukcije zasniva na principu ekvivalentnosti matrica, dok je skaliranje dinamičkog modela konstrukcije bazirano na Smith normalnoj formi. Skaliranje konstrukcije u operatorskom prostoru (matrice krutosti ili fleksije stavljaju se u međuodnos) trebalo bi biti bazirano na Sylvester matričnoj jednadžbi. Međutim, takav pristup nije praktičan te je zamijenjen Levenberg-Marquardt metodom za dobivanje približno ekvivalentnih matrica krutosti. Numerički primjeri ilustriraju predloženi drugačiji pristup.

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“…Throughout the workflow of the methodology, the estimation of unknown parameters is completed with a generalized Gauss-Markov least square, also called a weighted least square (WLS), procedure (Menke 2012;Tarantola & Valette 1982;Welsch et al 2000;Guillaume 2013). Based on the assumption that the observations are independent and Gaussian distributed, WLS provides the most likely solution (Caspary & Rüeger 1987;Welsch et al 2000;Guillaume 2013).…”

Section: Stochastical Integrationmentioning

confidence: 99%

“…The standardized residual errors allow a meticulous analysis of the adjustment and the detection of outliers. Moreover, this analysis is much more sensitive than the global chi-squared test, χ 2 (Menke 2012), because it allows each residual error value to be tested.…”

Section: Stochastical Integrationmentioning

confidence: 99%

Integrated velocity field from ground and satellite geodetic techniques: application to Arenal volcano

Muller¹,

Potro

Biggs³

et al. 2014

Geophysical Journal International

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S U M M A R YMeasurements of ground deformation can be used to identify and interpret geophysical processes occurring at volcanoes. Most studies rely on a single geodetic technique, or fit a geophysical model to the results of multiple geodetic techniques. Here we present a methodology that combines GPS, Total Station measurements and InSAR into a single reference frame to produce an integrated 3-D geodetic velocity surface without any prior geophysical assumptions. The methodology consists of five steps: design of the network, acquisition and processing of the data, spatial integration of the measurements, time series computation and finally the integration of spatial and temporal measurements. The most significant improvements of this method are (1) the reduction of the required field time, (2) the unambiguous detection of outliers, (3) an increased measurement accuracy and (4) the construction of a 3-D geodetic velocity field. We apply this methodology to ongoing motion on Arenal's western flank. Integration of multiple measurement techniques at Arenal volcano revealed a deformation field that is more complex than that described by individual geodetic techniques, yet remains consistent with previous studies. This approach can be applied to volcano monitoring worldwide and has the potential to be extended to incorporate other geodetic techniques and to study transient deformation.

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Continuous Inverse Theory and Tomography

Cited by 128 publications

References 6 publications

Localized Smart Interpretation - a data driven semi-automatic geological modelling method

Localized Smart Interpretation - a data driven semi-automatic geological modelling method

Similarity of structures based on matrix similarity

Integrated velocity field from ground and satellite geodetic techniques: application to Arenal volcano

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